Problem 46 - Chemistry The Science in Context

Question

For an electron in a hydrogen atom, how is the value of \(n\) of its orbit related to its energy?

Step-by-Step Solution

Answer

Answer: The relationship between the energy of an electron in a hydrogen atom and the principal quantum number "n" is given by the energy level formula: \( E_n = -\frac{13.6 \text{ eV}}{n^2} \). This formula shows an inverse square relationship, meaning that as "n" increases, the energy levels become less negative and the electron is located in orbits of higher energy, and vice versa.

Step 1: Understand the Energy Level Formula

The energy level formula calculates the energy associated with each orbit for an electron in a hydrogen atom. This formula depends on the principal quantum number "n" and the constant 13.6 eV, which is the ionization energy for a hydrogen atom. The energy levels are expressed in electron volts (eV), a unit of energy.

Step 2: Apply the Energy Level Formula

Use the energy level formula \( E_n = -\frac{13.6 \text{ eV}}{n^2} \) to determine the relation between the energy of an electron in a hydrogen atom and the value of "n". Remember that "n" represents the principal quantum number which indicates the electron's orbit: \( E_n = -\frac{13.6 \text{ eV}}{n^2} \)

Step 3: Analyze the Relation

The formula \( E_n = -\frac{13.6 \text{ eV}}{n^2} \) shows an inverse square relationship between the energy of an electron in a hydrogen atom and the principal quantum number "n". As "n" increases, the energy levels become less negative and the electron is located in orbits of higher energy. When "n" decreases, the energy levels become more negative, meaning the electron is located in orbits of lower energy. This relationship is important as it determines the energy required for an electron to jump between orbits and the emitted or absorbed energy in the form of photons.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by "n," is a critical component in quantum mechanics, particularly when studying atoms like hydrogen. It helps identify the energy levels in an atom and tells us about the electron's orbit or shell.
Each principal quantum number indicates a different energy level:

• "n" is always a positive integer (1, 2, 3,...).
• Higher values of "n" mean higher energy levels and larger orbits.
• When an electron is in lower energy level, such as n=1, it is closer to the nucleus compared to when "n" is larger.

The concept of the principal quantum number is crucial, as it determines the size of the electron's orbit, the energy it has, and influences the chemical properties of atoms and molecules. Learning these numbers enables us to understand more about the atom's structure and how electrons behave around the nucleus.

Energy Level Formula

The energy level formula for a hydrogen atom allows us to compute the energy of an electron's orbit. This formula is given by:\[ E_n = -\frac{13.6 \text{ eV}}{n^2} \] This equation is foundational in quantum physics and reflects the following key details:

• "E_n" represents the energy level of the electron.
• The constant 13.6 eV is the ionization energy for a hydrogen atom, representing the energy required to remove an electron from the lowest energy orbit.
• "n" is the principal quantum number discussed earlier.

Analyzing this formula shows an inverse square relationship. As "n" increases, the energy levels become less negative, indicating the electron is in higher energy orbits further from the nucleus. Conversely, a smaller "n" corresponds to lower energy orbits.

Electron Volts

Electron volts (eV) are a unit of energy commonly used in atomic and particle physics. They quantify the energy of electrons, making them ideal for studying energy levels in atoms.
To understand electron volts, think about:

• 1 electron volt is defined as the amount of kinetic energy gained by a single electron as it accelerates through an electric potential difference of 1 volt.
• eV simplifies the representation of minute energy measurements, as opposed to joules, which are often too large for atomic scales.
• In the case of the hydrogen atom, energy levels are measured in eV, like the 13.6 eV ionization energy mentioned earlier.

By using eV, physicists and chemists can easily calculate and interpret the energies associated with electrons in atoms, aiding in the study of atomic behavior and interactions.